Question: You decide you're only going to buy a lottery ticket if your expected winnings are larger than the ticket price. Tickets cost $$2$, and you get $$500$ if you win. The odds of winning are $1$ in $300$, meaning that you will win with probability $\dfrac{1}{300}$. Should you buy a ticket for this lottery?
Solution: The expected value of an event (like buying a lottery ticket) is the average of the values of each outcome. In this case, the outcome where you win is much less likely than the outcome that you lose. So, to get an accurate idea of how much money you expect to win or lose, we have to take an average weighted by the probability of each outcome. This means the expected value, considering both the price of the ticket and the possible winnings is $E = $ (money gained when you win) $\cdot$ (probability of winning) $+$ (money gained when you lose) $\cdot$ (probability of losing). Let's figure out each of these terms one at a time. The money you gain when you win is $$500$ and from the question, we know the probability of winning is $\dfrac{1}{300}$ When you lose, you gain no money, or $$0$ , and the probability of losing is $1 - \dfrac{1}{300}$ Putting it all together, the expected value is $E = ($500) (\dfrac{1}{300}) + ($0) (1 - \dfrac{1}{300}) = $ \dfrac{500}{300} = $\dfrac{5}{3}$ $$\dfrac{5}{3} - $2$ is negative. So, we expect to lose money by buying a lottery ticket, because the expected value is negative.